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Wednesday, May 16, 2012

Thermodynamics

Thermodynamics is the branch of science which deals
with energy (heat) transfer and its effect on state or condition of a system.
Essentially thermodynamics pertains to the study of:

1.Interaction of system & surrounding:-

It
relates the changes which the system undergoes to the influences to which it is
put.

2.Energy & its transformation:-

Energy inter-conversion in the form
of heat & work. James Joule had proved with his well known experiment that mechanical
work can be converted into heat energy. The credit for using heat by converting into work goes to James Watt who produced the first "Steam Engine"
& paved the way for industrial revolution.

3.Relationship between heat, work & physical
properties.

Such as pressure, volume & temperature of the
working substance (water for steam engine, petrol for bike etc) employed to
obtain energy conversion.

Thermodynamics has been excellently defined as the
science of three "Es" namely Energy, Entropy & Equilibrium. The
principles & concepts of thermodynamics are important tools in the
innovation, design, development & improvement of engineering processes,
equipment & devices which deals with effective utilization of energy.

The applications of engineering thermodynamics in the field
of energy technology are as follows:-

3.Chemical process plants & direct energy
conversion devices. A large number of processes in various fields such as
agriculture, textiles, dairy, drugs & pharmaceutical industries are also
governed by thermodynamics principle.

Macroscopic & Microscopic approach.

or Classical & Statistical Thermodynamics.

There are two approaches for investigating the
behavior of a system.

Macroscopic approach.

It is also known as
classical thermodynamics which is concerned with gross or overall behavior of
matter. In classical thermodynamics ,the
analysis of thermodynamics systems is
explained with the measurable property like pressure ,volume, temperature etc. It doesn’t explain the structure of
matter.(i.e No attention is focused on the behavior of individual particles constituting the matter.)The volume considered
is very large compared to molecular dimension & the system is regarded as
bulk (whole/single).The study is made of overall effect of several molecules.
The behavior & activities of the molecules are averaged. Only a few
variables are needed to describe the state or condition of matter. The
properties like pressure, temperature etc. needed to describe the system can be
easily measured & felt by our
senses. The particles of system required simple mathematical formula for
analyzing the system for e.g. Piston cylinder assembly of an IC engine, volume
occupied by the gas for each position of piston etc. It is based on the concept
of continuum. The macroscopic form of energy are those a system possess as a whole
with respect to some outside reference frame, such as potential and kinetic
energies.

Microscopic point of view

This is also known as statistical thermodynamics and is
directly concerned with structure of matter. It focuses on statistical behavior
of mass consisting numerous individual molecules and correlates macroscopic
properties of the matter with molecular configuration and intermolecular
forces. It explains the structure of matter. Few coordinates are not sufficient
to describe this system. Concept of
continuum is not valid for this viewpoint. The microscopic form of energy are
those related to the molecular structure of a system and the degree of
molecular activity, and they are independent of outside reference frames. The sum
of all the microscopic forms of energy is called the internal energy of a
system.

Thermodynamics system:-

It is defines as a definite area or a space where some
thermodynamic process takes place. It is a region where thermodynamic process are
studied. A thermodynamic system has it’s boundary and anything outside the boundaries is called it’s
surrounding. These boundaries may be fixed like that of a tank enclosing a
certain mass of compressed gas, or movable like boundary of a certain volume of
liquid in a pipe line.

Thermodynamic system
represents a prescribed & fixed quantity of matter under consideration to
analyze a problem; to study the changes in its properties due to exchange of
energy in the form of heat & work. The system may be quantity of steam, a mixture
of vapor & gas or a piston cylinder assembly of an IC engine & its
contents.

For the description of thermodynamic system some of
the following quantities need to be specified.

1.Quantity as well as composition of matter.

2.Measurable properties such as pressure,
temperature & volume of the system

3.Energy of the system

The combination of matter & space, external to
the system that may be influenced by the changes in the system is called
surrounding or environment. The thermodynamic system & surrounding are separated
by an envelope called boundary of the system. The boundary represents the limit
of the system and may be either real or imaginary and may change shape, volume,
position or orientation relative to observer for e.g. an elastic balloon which
is initially spherical in shape may change into cylindrical shape or some other
geometrical shape or may get squeezed to reduce volume during a certain period.
As such, the boundary of the gas in the balloon couldnot retains the same size
and shape. Further, the boundary may be diathermal or adiabatic depending upon
whether it allows or not exchange of energy in the form of heat. The walls of
the boundary which does not let heat transfer to take place across them are
named as adiabatic. In contrast the walls that do allow heat interaction across
them are called diathermic. The diathermic system can be classified into.

A.Closed system

A close system can change energy in the form of heat & work with
its environment but there is no mass transfer across the system boundary. The
mass within the system remains the same and constant, though its volume can
change against flexible boundary further the physical nature & chemical
composition of the mass may change. Thus, a liquid may evaporate, a gas may
condense or a chemical reaction may occur between two or more components of the
system for e.g.

Cylinder fitted
with movable piston.

1.Motor car battery.

2.Pressure cooker

3.Refrigerator

4.Ice-cream greaser

5.Bomb Calorimeter etc.

B.Open system:

A open system has mass exchanged with the surrounding along with
transfer of energy in the form of heat and work. The mass within the system
doesn't necessarily remain constant. It may change depending upon mass inflow
& mass outflow for e.g.

1.Water wheel.

2.Motorcar engine:-

The engine initially draws charge
(mixture of air and petrol) and finally exhausts the spent up gases to the
surrounding atmosphere. The mass flows across the engine boundary.

3.Steam generator (Boiler):-

A boiler is a device which converts
water from liquid vapor freeze. Here consist of system change, water froze into
and steam froze out of the system.

Most of the
engineering devices are open system.

C.Adiabatic system:

There exists wall or boundary
which doesn't allow heat transfer to take place across them, no matter how not
one member is compared to other. Such boundaries are named as adiabatic. An
adiabatic system is thermally insulated from its environments. It is enclosed
by adiabatic walls and can exchange energy in the form of work only for e.g. a
pipe carrying heat enclosed by thermal insulation.

D.Isolated
system:

An isolated system is a
fixed mass and energy; it exchanges neither mass nor energy with another system
or with surrounding. An isolated system has no interaction with the
surrounding, it neither influences the surrounding, nor it is influenced by it.
When a system and its surrounding are taken together they constitute an
isolated system. The universe can be considered as an isolated system & so
is the fluid enclosed in the perfectly insulated closed vessel (thermos/flask)

E.Homogeneous & heterogeneous:

A system consisting of single phase is called homogeneous
system for e.g.

1.Ice, water, dry saturated steam.

2.A mixture of air and water vapor

3.Mixture of ammonia in water

A system whose mass
content is non uniform through i.e. it consists of more than one phase is
called heterogeneous system for e.g.

1.A mixture of ice and water

2.A mixture of water & gasoline

3.A mixture of water & mercury

Thermodynamic property

Thermodynamic
property refer to the characteristic which can be used to describe the
condition or state of the system for e.g. temperature, pressure, chemical
composition, color, volume, energy etc. The salient aspects of thermodynamic
properties are:-

a)It is a measurable characteristic describing
system & helps to distinguish one system to another

b)It has a definite unit value when the system is
in a particular state

c)It is dependent only on the state of the system,
it does not depend on path or route. The system follows to attain the
particular state.

d)It's differential is exact

If 'P' is the
thermodynamic property with dP representing its differential change, then the
integral between initial state 1 to final state 2 of system will have only one
value given by 'O2'

Since, the thermodynamic
property is the state of a system; it is referred to as point a function or
state function. There are two kinds of thermodynamic property namely Intensive
and extensive.

a)Intensive Property:

It is independent of the extent or the mass
of the system. Its value remains same whether consider whole the system or part
on it for e.g. pressure, temperature, density, viscosity, composition, thermal
conductivity, electrical potential etc.

b) Extensive property:

It depends on the mass or
extent of the system. Its value depend on how big portion of the system is
being considered for e.g. energy, enthalpy (heat contained), entropy (disorder),
volume etc.

Specific property:-

An extensive property expressed per unit mass of the
system is known as specific property for e.g. specific volume, specific energy,
specific entropy, specific enthalpy etc.

Equilibrium:-

Equilibrium is the concept associated with the
concept of tendency for spontaneous change in the value of any macroscopic
property of the system when it is isolated from its surrounding.

1.
Mechanical equilibrium (Equality of pressure)

Condition or state in which
there is no unbalance force within the system and nor its boundaries.
Mechanical equilibrium implies uniformity of pressure i.e. there is one value
of pressure for the entire system; diffusion takes place to wipe out the
unbalance and attain the state of mechanical equilibrium.

2.Chemical Equilibrium:-

A system
in chemical equilibrium may undergo a spontaneous change of internal structure
due to chemical reaction or diffusion (transfer of matter) for e.g. there will
be spontaneous change in the properties of the mixture of oxygen & gasoline
once it is ignited. Chemical equilibrium represents that condition a state of
the system when all chemical reaction in it have ceased & there is no mass
diffusion.

3.Thermal Equilibrium

A system is said to be in thermal equilibrium, when is no
temperature difference between the parts of the system or between the system
and surroundings.

4.Thermodynamic Equilibrium

A system
which is simultaneously in a state of mechanical equilibrium, chemical
equilibrium & thermal equilibrium is said to be thermodynamic equilibrium.

⋆ State, path, process cycle.

Consider a system constituting by a
gas enclosed in a piston cylinder assembly of reciprocating machine.
Corresponding to the position of the piston at any instant, the condition of
the system will be prescribed by pressure, volume & temperature of the gas.
When all such properties have a definite value, the system is said to be exist
at definite state. State is thus condition of the system identified by
thermodynamic property.

When a piston move outwards the properties of the system change
(pressure decreases, volume increases). Any such operation in which properties
of the system change is called change of state. The locus of series of states
through which a system passes is going from initial state to final state
constitutes the path.

A complete specification of the
path is referred to as process.

When
a system in a given state undergoes series of processes such that the final
state is identical with initial state, a cyclic process or a cycle is said to
be be executed. The cycle 1-A-2-B-1 consist of 1-A-2 & 2-B-1(process),
cycle 1-C-2-B-1 consist of 1-C-2 & 2-b-1 (process).

Quasi-state or Quasi Equilibrium process

Some unbalanced
potential must exist either within the system or between a system & its
surrounding to promote the change of state during a thermodynamic process.
Consider a system of the gas in cylinder fitted with piston upon which are placed many small pieces of
weights.

The
upward force exerted by the gas just balances the weight on the piston. The
system is initially in equilibrium in equilibrium state indentified by pressure
P1, Volume V1 & Temperature T1. When these
weights are removed slowly one at a time, the unbalanced potential is
infinitesimally small. The piston will slowly move upward & at particular
instant of piston travel, the system would be almost closed to the state of
equilibrium. The departure of system
from thermodynamic equilibrium state will be infinitesimal small. Every state
passed by the system will be in equilibrium state. The locus of series of such equilibrium state
is called Quasi-static or Quasi Equilibrium process.

Thus, when the process is carried out in such a way that
at every instant, the system deviation from the thermodynamic equilibrium is
infinitesimal, then the process is known as quasi-static or quasi equilibrium
process. It is a very slow process and each state in the process may be
considered as an equilibrium state. It is also known as reversible process.

PERIODIC MOTION:-

The motion which
repeats after a regular interval of time is called periodic motion. In such a
motion the particle always come to rest is called mean or equilibrium position.
If the particle is displaced from mean position there exist certain kind of
force which tries the particle back to its mean position, such a force is
called restoring force. Restoring force is the function of displacement.
Periodic motion can be expressed in terms of trigonometric function. So, it is
called harmonic motion.

SIMPLE HARMONIC MOTION

Simple harmonic motion is the
special kind of oscillatory motion in which the body moves again and again over the same path about the
fixed point (generally called equilibrium position) in such a way that it is
acted upon by a restoring force ( or torque) propotional to it’s displacement (
or angular displacement) and directed towards the mean position.

SHM is a special
type of periodic motion in which particle oscillates in straight line in such a
way that acceleration is directly proportional to the displacement from mean
position & it is directed towards mean position.

SOME TERMS ASSOCIATED WITH S.H.M.

1.Amplitude:-

The
maximum displacement of the particle from mean position is called amplitude of
the oscillation.

As
the particle displacement is, x=Asin(ωt+δ) and sin(ωt+δ) varies from -1 to +1

∴ Maximum
displacement =±A

A
is called amplitude of oscillation.

2.Time period:-

The time required to complete
one oscillation is called time period. It is denoted by T.

We have,
displacement of particle in SHM as, , x=Asin(ωt+δ)…(i)

The position of
particle will be same after time T i.e.
x=Asin[ω(T+t)+δ]….(ii)

∴ sin(ωt+δ) repeats
its value every 2π or even integral of π.

∴ ω(T+t)+δ =
ωt+δ+2π

or, ωT=2π

This is the time
period of S.H.M.

3.Frequency:-

The
number of oscillation in one second is called frequency. It is denoted by 'f'
and given as,

4.Angular frequency (ω):-

5.Phase & phase constant:-

Phase
is the status of the particle which executes S.H.M.

We have,
x=Asin(ωt+δ) and v=Aωcos(ωt+δ)

(ωt+δ) is responsible for the status of particle therefore
it is phase of the particle. It is denoted by ϕ.

Phase (ϕ)=ωt+δ

The phase increases with time.

δ is called phase constant which depends upon the choice of
instant time (t=0). If we chose the instant time as origin then, ωt+δ=0 ∴ δ=0

This
means phase constant will be zero.

OSCILLATION OF SPRING MASS SYSTEM

Consider a
spring of length 'l' which is supported by a fixed rigid support at one end and
another end is free. Let 'm' be the mass attached to free end. Due to mass 'm'
let 'l' be the elongation produced then, according to Hook's law, the force
applied is directly proportional to the extension produced,

i.e. F1∝
l

or, F1=-kl…(i)

-ve sign indicates
that the force is restoring.

Now, further pull
the mass through a distance 'x' then, the force will be,

F2=-k(l+x)…..(ii)

∴ the net force
applied so that the spring mass system sets into oscillation is,

F=F2-F1

or, F=-k(l+x)-(-kl)

or, F=-kx….(iii)

According to
Newton;s law,

F=ma

from (iii) & (iv)

which is equation of S.H.M i.e. the spring mass system
executes S.H.M.

This
is time period of spring mass system.

ANGULAR HARMONIC MOTION

Angular harmonic motion is the periodic
motion in which angular acceleration is directly proportional to angular
displacement & it is always acting towards mean position.

If θ be the displacement & α
be the angular acceleration, then,

α ∝ θ

or,
α =-ω²θ

COMPOUND PENDULUM

Limitation of simple pendulum

a.The point mass heavy bob is not possible.

b.The mass less inextensible spring is not
possible.

c.In simple pendulum the centre of oscillation
& centre of gravity lies at the same point which is practically not
possible

d.Since the string has certain mass, it has
certain moment of inertia which is not considered in it.

e.It is an ideal pendulum.

Compound
pendulum is a rigid body of any shape capable of oscillating in horizontal axis
in vertical plane not passing through centre of gravity.

Let
'mg' be the weight acting vertically downward through c.g. Let 'l' be the
length of compound pendulum i.e. distance from centre of suspension to centre
of gravity(G). Now, displace the pendulum through angle 'θ' & G' be the new
position of c.g. , then the force at G'and its reaction at c.s constitutes
couple.

Then,
torque is given as,

Torque (τ)=Force × perpendicular
distance from axis of rotation

or τ = mg × lsinθ

If
the displacement is small then,

sinθ
≃ θ

∴
τ=mglθ

which
provides restoring force i.e. which tries the pendulum back to its mean
position.

∴ Restoring torque (τ)=
-mglθ………(i)

If I be the moment of inertia
and α be the angular acceleration, then,

The amount of work
done to displace a particle through distance dx will be=> dW=F× dx….(ii)

from (i) &(ii)
, dW=-kx × dx

The total work done
to displace x distance is,

∫ dW = ∫-Kxdx

This amount of work
done is changed into potential energy.

If 'm' be the mass & 'v' be the velocity of particle executing
S.H.M. then,

but, x = A.sin(ωt+δ)

y=A.ω.cos(ωt+δ)

, which is
independent of time i.e. total energy remains conserved.

NUMERICAL

Q1. 2 Kg. mass hang
from a spring. A 300g body hang below the mass stretches the spring 2 cm
further. If the 300g body is removed and the mass set onto oscillation. Find
the period of motion.

F=-Kx

or, F=Kx (for
magnitude only)

or, mg=Kx

=150 N/m

Q2. A harmonic
oscillator has a period of 0.5 sec. & amplitude of 10 cm. Find the speed of
object when it passes through the equilibrium position.

Vmax = A.ω

= A× 2πf

Q3. A body of mass
0.3 Kg executes S.H.M. with a period of 25 sec. and amplitude of 4 cm.
Calculate the max. velocity, acceleration & K.E.

Soln.

m= 0.3 Kg, T=25
sec., A=4cm .

Vmax=A.ω

amax=Aω²

Q4. A small body of
mass 0.1 Kg is undergoing S.H.M. of amplitude 1.0 m and period 0.2 sec (a) what
is the maximum value of force acting on it?, (b) If the oscillations are
produced by a spring, what is the constant of spring?

Fmax=m.Qmax

Fmax=m.Aω²

Q5. Two springs
having force constants k1 & k2 respectively are attached to a mass and two
fixed supports as shown. If the surfaces are frictionless, find the frequency
of oscillation.

Soln.

Total restoring force,

F = -k1x-k2x

F = -k1x-k2x

ma = -(k1+k2)x

Q6. Show that if a uniform stick of length 'l' is mounted so as to
rotate about a horizontal axis perpendicular to the stick and at a distance'd'
from the centre of mass. The period has a minimum value when d=0.289 l.

Q8. At a time when the displacement
is half the amplitude what fraction of the total energy is kinetic & what
fraction is potential energy. in S.H.M ?At what displacement is the energy half
K.E. & half P.E.

Q9. Uniform circular disc of radius
R oscillates in a vertical pane about horizontal axis. Find the distance of the
axis of rotation from the centre of which the period is minimum. What is the
value of the period?

Soln.

Q10. A particle
executing S.H.M. of amplitude 'A' along the X-axis. At t=0, the position of the
particle is and it moves
along the positive X-direction. Find the phase constant δ. If the equation is
written as x=sin(ωt+δ)

WAVE MOTION

Introduction

The disturbance which travels
onward through the medium due to the periodic motion of particle about their
mean position is called wave motion. Thus wave transfers energy from one place
to another place without any bulk transformation of intervening particles of
the medium, so wave is the mode of transfer of energy.

According to the medium required
or not required wave is categorized into two types.

Mechanical wave

The wave which require material medium to travel or propagate onward is
called mechanical wave for e.g. sound wave.

Electromagnetic wave

The wave which do not require material
medium to travel or propagate onward is called electromagnetic wave for e.g.
radio wave, light wave etc.

According to the vibrations of particles
wave motion is of two types

Transverse wave motion:-

The wave motion in which particles of
medium oscillates in perpendicularly to the propagation of wave is called
transverse wave motion. Therefore it is in the form of crest and trough.

Longitudinal wave motion:

The
wave motion in which particles of the medium vibrates parallel to the direction
of propagation of wave is called longitudinal
wave motion. Therefore it is in the form of compression & rarefraction.
It is also called as Pressure wave.

Progressive wave:

The wave which
travels onward through the medium in a given direction without alternation or
with constant amplitude is called progressive
wave. It may be transverse or longitudinal wave. In such wave all particles
vibrate with same amplitude, the only difference is that they have different
instant of time.

Consider a particle at 'O' executes
periodic motion then, if 'y' be the displacement of particle then equation for
displacement,

y=Asinωt…..(i)

Let 'P' be the
point at a distance 'x' from origin,

In one complete
oscillation the phase is 2π or wave covers the distance λ.

when, path
difference is λ then phase difference is 2π

when path
difference is 1 then phase difference is 2π/λ

when path
difference is x then phase difference is

∴
path difference between O & P will be

If the wave travels
in –ve X direction then,

y=Asin (ωt+kx)

Wave velocity & particle
velocity

The rate of change of wave
displacement w.r.t. time is called wave velocity (phase velocity). It is
denoted by 'u' and given by,

We have wave
equation for progressive wave travelling in +ve direction,

y=Asin(ωt-kx)……(ii)

Now,
differentiating w.r.t. time we get,

For given wave
ωt-kx=constant

Now,
differentiating w.r.t. time we get,

∴
wave velocity = wavelength × frequency

The rate of change
of displacement of the particle with time is called particle velocity. It is
denoted by 'v' & given by,

but, y=Asin(ωt-k)
….(iv)

Differentiating
w.r.t. x

From eqn. (v) & (vi) we get,

Particle velocity =
wave velocity × (-slope displacement curve)

Intensity of wave:-

The flow of energy per unit area
per unit time is called intensity of wave.

Consider
a table of cross section area's' and length 'l' when the wave energy is transmitted. Let 'n' be the number of
particles per unit volume then the total number of particle in volume 'sl' will
be 'nsl'.

Hence, Intensity ∝ ρ

Hence, Intensity ∝ u

Hence, Intensity ∝ f²

Hence, Intensity ∝ A²

VELOCITY OF TRANSVERSE WAVE ALONG
STRETCHED STRING.

The
velocity of transverse wave
along stretched string can be expressed in terms of tension along the string
& mass per unit length of the string.

When
a jerk is given to the stretched string fixed at one end transverse wave can be
produced.

Consider
an element length Δl of string which forms an arc 'PQ' of radius 'R'. The
tension produced on the string can be resolved into two perpendicular
components. Tcosθ along the horizontal & Tsinθ along radial direction. The
horizontal components cancel each other and the radial components provide the
tension.

The
resultant tension along string = Tsinθ +Tsinθ =2 Tsinθ…(i)

Energy Transmission along
stretched string

If μ be the mass per
unit length (m/Δl) of string then, m=μΔ L

STANDING OR
STATIONARY WAVE

When two travelling waves of
same amplitude & same wavelength travel in opposite direction with same
velocity superimpose with each other then the resultant wave formed is called standing wave or stationary wave.

There are certain
position like 'N' where the particle vibration is minimum i.e. particle doesn't
vibrate is called node. There are
certain positions like 'AN' where the particle vibration is maximum is called Anti-Node.

Q5.Two waves are simultaneously passing through a
string. The equation of waves are y1=A1sink(x-vt), y2=
A2sink(x-vt+x0), where the wave no
k=6.28 cm-1 & x0=1.50 cm, the amplitudes are A1=50mm
& A2=40mm.

WAVE
OPTICS (PHYSICAL OPTICS)

GEOMETRICAL PATH & OPTICAL PATH

The distance by the light
through a certain distance for a certain time in a certain medium is called
geometrical path.

The distance travelled by the
light in vacuum through a certain distance for the same time in which it travel
in medium is called optical path.

Suppose light travels through
the medium of refractive index 'μ' with velocity 'v' for time't' then the
geometrical path will be,

x=ut

If 'c' be the velocity of light
in vacuum for time 't' then distance travelled by light,

Interference of light waves

The
source of light distribute its energy uniformly in all directions if there are
two sources which distributes energy of same wavelength & frequency the
energy distribution will not be uniform. The non uniform distribution of energy
due to the superposition of two light waves of same amplitude and frequency is
called interference of light waves.

At
some point energy will be maximum called constructive
interference & at some places energy will be minimum called destructive interference.

Coherent source

The sources are said to be
coherent if they produce the light wave of same amplitude & frequency.

For the sustain
interference following conditions are to be satisfied:-

1)Two sources should be coherent

2)Two sources emit light wave continuously.

3)They should be narrow sources

4)Sources should be close to each other

5)The distance between the sources and screen
should be far

6)Sources should be monochromatic

Condition for interference maxima & minima

Consider
two point sources S1 & S2 which produces the interference fringes. Let 'P'
be the point where we have to find the interference fringe, which is either
maxima or minima.

Let
y1 = asinωt be the wave produced by S1 & y2= asin(ωt+δ)
be the wave produced by S2.

The resultant displacement of waves
by superposition principle can be written as,

y = y1+y2

= asinωt+
asin(ωt+δ)

= a[sinωt+
sinωt.cosδ+cosωt.sinδ]

= A[sinωt(1+cosδ)+cosωt.sinδ]….(i)

Let a(1+cosδ) = Acosθ
…..(ii)

asinδ = Asinθ
…….(iii)

then,
y = A[sinωt.cosθ+cosωt.sinθ]

= Asin(ωt+θ)….(iv)

,
which is the resultant wave equation of the resultant wave.

Squaring
& adding (ii) & (iv)

a²sin²δ+a²(1+cosδ)²=A²(sin²θ+cos²θ)

a²sin²δ+a²+a²cosδ+ a²cos² δ=A²

or, 2a²+2 a²cosδ=A²

or, 2a²(1+cosδ)=A²

since, intensity of the wave is
directly proportional to the square of amplitude, we can write,

The
intensity will be maximum if =1

for 2π phase difference he path difference will be λ

The point 'P' will correspond to maximum or bright fringe if
the path diff = nλ

for 2π phase difference path difference = λ

Thin films:

Thin
films is an optical medium where thickness is order of wavelength of light in visible region (400 nm
t 760 nm). It may be the soap bubble, glass or air enclosed between two
transparent glass plate whose thickness is 0.5μ m to 10μm.

*Interference due to thin films

*Interference due to reflection on thin film

Let AB be
the incident light incident in upper surface, XY of the thin film of
thickness't' and refractive index 'μ '. At point B, most of the part
transmitted along BC and a point of it is reflected along BF.

Also, at point 'C', most part is transmitted along CN and a
point of it is reflected along CD. Similarly at point 'D' most part transmitted
along DC and a part of it is reflected along 'DE'.

Only the
light wave reflected from point 'B' and 'C' are of appreciable strength.

Now, the path difference between BF and DG = BC +CD-BN

Optical path difference = μ(BC+CD)-BN…(i)

Now, from fig. in Δ BCM,

from, (iv) & (v)

BN = 2tanr.sini….(vi)

At 'B' reflection takes
place from the surface of denser medium, therefore the additional path
difference λ/2 should be added [At
reflection the wave losses half of the wave]

for minima or dark
fringe,

path difference = (2n+1)
λ/2 [n=0,1,2,….]

Newton's Ring:

Let 'S' be the source of
monochromatic light. Light rays form 'S' are made parallel by lens L. These
parallel rays are incident on glass plate 'G', kept 45° with horizontal. A part
of these rays fall normally on plane convex lens 'P' kept on glass plate 'AB'.
The Plano-convex lens and glass plate AB
create air film of variable thickness.

The light ray reflected form
upper part of thin air film and lower part of thin film interfere to produce
interference fringes as from circle(ring). Newton's rings are the fringes of
equal thickness.

Let 'R' be the
radius of curvature of Plano-convex lens, 't' be the thickness of air film for
which radius of nth ring will be 'rn'.

For the interfere due to
reflection on thin film, the path diff=2μtr+λ /2

For air film, μ = 1

for normal
incidence, r=0

∴ path diff=2t+λ /2

for bright fringe path difference = nλ

∴ 2t+λ /2 = nλ [n=0,1,2,….]

2t=nλ -λ /2

2t = λ/2
(2n-1)….(i)

for dark fringe,
path diff = ((2n+1) λ /2

∴ 2t+λ/2 = (2n+1) λ /2

2t = nλ

from fig, in Δ PNM

PM²
=PN² +NM²

R²=rn²
+(R-t)²

R²=rn² +
R²-2Rt+t²

rn²=2Rt-t²

but, the radius of curvature is
very greater then thickness of air film.

→2RT≫t²

→rn²
=2RT

→2t= rn²/R=Dn²
/R

Where, Dn=
diameter of an fringe.

for bright fringe,
eqn. (i) & (iii) must be equal

If dm be
the diameter of mthring then

Now, subtracting
(v) fro (iv)

If we know the diameter of nthand nth ring, we can determine the wavelength of monochromatic
light.

Diffraction:- It is the phenomenon of bending of light round
the sharp corner and spreading the light into the geometrical shadow region. As
a result of diffraction, maxima and minima of light intensities are found which
has unequal intensities. Diffraction is the result of superposing of waves,
from infinite of coherent sources on the same wave front, after the obstacle
has distorted the wave front.

There are two types of diffraction,

a.Fresnel diffraction :-

b.Fraunhoffer diffraction :-

TO observe the diffraction phenomenon the
size of the obstacle must be of the order of the wavelength of the waves.

Fresnel diffraction :- It involves
spherical wave fronts. If either source or screen or both are at a finite
distance from the diffracting device , the diffraction is called Fresnel
diffraction and the pattern is shadow of the diffracting device modified by
diffraction effects. Diffraction at a straight edge, narrow wire or small
opaque disc is familiar examples of this type. It explain diffraction in terms
of “half period zones” , the area of
which is independent of the order of zone.

Fraunhoffer diffraction :-

It is the diffraction when both source and
screen are effectively at infinite distance from the diffracting device and
pattern is the image of source modified by diffraction effects. Diffraction at
single slit, double slit and diffraction grating can be cited as examples of
this type. It follows that fraunhoffer diffraction is an important special case
of Fresnel diffraction.

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